Partition interactions and sharp binary rigidity

Higher-order dependence depends on which partition-based signed interaction measure is used. At four variables, we express the Streitberg interaction exactly as the Lancaster interaction minus the three tensor products of paired dependence defects. Thus agreement on the pure-interaction stratum does not imply that the two operators coincide for arbitrary laws. For four balanced sign variables, the fourth joint cumulant satisfies the sharp inequality \(|\kappa_4|\le2\). We characterize all eight common-sign equality laws, and obtain an exact formula for total-variation distance to the full equality family through the relative-sign distribution. An exact local expansion yields the optimal leading stability coefficient \(1/4\), while an explicit family rules out a corresponding global linear bound. We additionally compare normalized partition interactions via lower-order marginal dependence defects. Classical partition and cumulant identities provide the setting; the contribution is the sharp balanced-sign extremal geometry, complete equality classification and local quantitative rigidity.